And there’s some evidence in last week’s otherwise unimpressive jobs numbers that it’s not an impossible mission.

The unemployment rate appears to be dipping for post 9/11 veterans, according to data from the Bureau of Labor Statistics. In the first six months of 2012, that figure has mostly stayed below 10 percent, in contrast to 2011, when the unemployment rate for that group hit double digits every month that year except for August.

These mostly younger veterans, known officially as Gulf War Era-II veterans, have experienced higher rates of unemployment than just about any other demographic in the country. According to last Friday’s figures, that number is now at 9.5 percent, down from last June’s 13.3 percent.

The trend is promising, although unemployment figures are volatile, said Rob Diamond, director for the Obama campaign’s veterans and military families outreach initiative.

“When you look at it over the last 12 months, veterans’ unemployment has fallen — but we still need to do more,” Diamond said. “The president has been laser-focused on the issues of veterans’ employment,” he added, citing initiatives like two tax credits enacted by Obama encouraging businesses to hire veterans.

Veterans, a traditionally Republican constituency, could play decisive roles in a number of key swing states this year. But it’s hardly a done deal that Obama will win them over. Obama has a better shot among younger veterans, just as he does with younger voters — while presumptive GOP nominee Mitt Romney is likely to win over older veterans in the same way he tends to appeal to older voters more generally.

Republicans are skeptical of Chicago’s efforts.

“If I were the Obama people, I would not be counting on young military veterans to be the deciding vote in Florida,” said Rick Wilson, a Tallahassee-based Republican strategist. “These guys, once they’re out of the service, they face the same kinds of terrible economic challenges that other folks are encountering in the Obama economy.”

Wilson also added that in places like Florida, with its 1.6 million veterans, the average former service member tends to be older.

Older veterans are often male, white and conservative — and likely to vote. That dynamic favors Romney.

According to a Gallup poll from late May, veterans as a whole still overwhelmingly opt for the Republican choice: Romney was favored by a margin of 58 to 34 percent in that survey. His campaign attributes that both to the poor economy — which, it argues, Romney can turn around — and Obama’s willingness to cut the defense budget.

“There’s a clear contrast on the economic vision of both candidates,” said Romney spokesman Ryan Williams. “There are also clear differences on how to fund and maintain our military. The president has put forth a plan that proposes drastic cuts to the armed forces, and Romney opposes that. It’s a bright line that divides the two candidates.”

Last week, Romney’s team unveiled a Veterans and Military Families for Romney Coalition, led by Sen. John McCain, former President George H.W. Bush, former Senate majority leader and Republican presidential candidate Bob Dole, and a host of other political and military leaders.

“The thing to keep in mind about veterans, in particular among younger veterans is, unemployment is extremely high,” said Joshua Baca, the national coalitions director for the Romney campaign. He added, “At the end of the day, this election is going to be about the economy… that’s going to be a very significant message that we push forward: that our veterans, who honorably serve our country, are coming home to a lack of opportunity.”

But the Obama campaign thinks the president’s record on veterans’ issues speaks for itself. Chicago is highlighting the president’s achievements — touting policies like tax credits and bringing troops home from Iraq and the extension of GI bill benefits to Afghanistan and Iraq veterans — in places like Colorado, Virginia, North Carolina, Nevada and Florida, all of which are also critical swing states.

Obama’s veterans’ outreach is part of Operation Vote, a grassroots strategy that targets specific demographics — including African-Americans and women. The Veterans and Military Families for Obama group held kick-off events earlier this year in places like Norfolk, Va. and Fayetteville, N.C., and is focusing campaign events in states with larger veterans’ populations.

Florida, for example, has the third-highest population of veterans in the country, according to Steve Murray, a spokesman for the Florida Department of Veterans’ Affairs. While most of them do tend to be older, a still-sizeable 231,000 veterans of the wars in Iraq and Afghanistan, who are often young, call the Sunshine State home.

The age demographics are similar in Colorado, said Jason Crow, a veteran of Iraq and Afghanistan who advises the president’s Colorado campaign on military and veterans’ issues. He estimated that about 24,000 veterans — most of them young — are receiving education benefits of some sort from the Veterans’ Administration through Obama’s post-9/11 GI bill, though the state as a whole has more than 400,000 veterans, he said.

“What we’re seeing is a shifting trend of younger veterans being more open to Democratic candidates and Democratic ideals,” Crow said. “Our veterans are coming back after three, four deployments, and they see President Obama kept his promise to responsibly end the war in Iraq and he’s starting to do the same in Afghanistan…Younger veterans recognize and appreciate his responsible approach to military and national security policy, and they are supporting him for that as much as for education, health care, employment issues.”

Last time around, Obama lost the veterans’ vote by a margin of 55 to 45 percent to McCain, a Vietnam War hero. But Obama’s team is quick to point out that the president won with veterans under the age of 60.

“What’s been illustrated over the last three-and a half years is, this is a president that understands the community, that understands the challenges this community faces,” said Clo Ewing, deputy press secretary for the Obama campaign.